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\begin{document}

\title{A Practical Formulation for an Anisotropic and Nonstationary Mat\selectlanguage{ngerman}érn
Class Correlation Operator\selectlanguage{english}}



\author[1]{Timothy A Smith}%
\affil[1]{Cooperative Institute for Research in Environmental Sciences (CIRES) at the University of Colorado Boulder}%


\vspace{-1em}



  
  \date{May 19, 2024}


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\begin{abstract}
A key component of data assimilation methods is the specification of
univariate spatial correlations, which appear in the background-error
covariance. For realistic problems in meteorology and oceanography,
correlation length scales are nonstationary (variable in space) and
anisotropic (variable in each direction). Variational approaches
typically use an operator to enforce correlation length scales, and thus
the operator must be designed to capture desired levels of
nonstationarity and anisotropy. For systems with complex boundaries,
such as the ocean, it is natural to use a filtering approach based on
the application of an elliptic, Laplacian-like operator. Here we show
how an elliptic operator can be formulated to capture a general
Mat\selectlanguage{ngerman}érn-type correlation structure. We show how nonstationarity and
anisotropy can be encoded into the operator via a simple change of
variables based on user-defined normalization length scales. The change
of variables defines a mapping between the computational domain and a
space where the analytical Matérn correlation function applies. In
addition to the mapping, two other hyperparameters separately control
the correlation length scale (i.e. range) and shape. As a practical
use-case, we apply the operator to a global ocean model. We show that
when the normalizing length scales correspond to the local grid scale,
the range parameter has an intuitive interpretation as the number of
neighboring grid cells at which correlation drops to 0.14. Finally, the
correlation model is shown to be computationally efficient in two
regards. First, the necessary linear solve can be performed with a high
tolerance (\selectlanguage{english}~10\^{}\{-3\}) while still achieving the correct statistics,
requiring few iterations to converge. Secondly, the operator's exponent,
which controls the correlation shape, is linearly related to the
diagonal elements of its matrix representation. As a result, using an
exponent greater than one can improve convergence properties. Thus, the
framework provides flexibility in controlling correlation shape.%
\end{abstract}\selectlanguage{ngerman}%



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